Stochastic Comparisons and Dependence of Spacings from Two Samples of Exponential Random Variables

Let X ₁,X ₂,…,X _n be independent exponential random variables such that X _i has hazard rate λ for i = 1,…,p and X _j has hazard rate λ* for j = p + 1,…,n, where 1 ≤ p < n. Denote by D _i:n (λ, λ*) = X _i:n  ? X _i?1:n the ith spacing of the order statistics X _1:n  ≤ X _2:n  ≤ ··· ≤ X _n:n , i = 1,…,n, where X _0:n ≡ 0. It is shown that the spacings (D _1,n ,D _2,n ,…,D _n:n ) are MTP₂, strengthening one result of Khaledi and Kochar (2000), and that (D _1:n (λ₂, λ*),…,D _n:n (λ₂, λ*)) ≤ _lr (D _1:n (λ₁, λ*),…,D _n:n (λ₁, λ*)) for λ₁ ≤ λ* ≤ λ₂, where ≤ _lr denotes the multivariate likelihood ratio order. A counterexample is also given to show that this comparison result is in general not true for λ* < λ₁ < λ₂.     

Dependence Structure of Spacings of Order Statistics

Let X_1:n ≤ X_2:n ≤···≤ X_n:n denote the order statistics of a sample of n independent random variables X₁, X₂,…, X_n, all identically distributed as some X. It is shown that if X has a log-convex [log-concave] density function, then the general spacing vector (X_k1:n, X_k2:n ? X_k1:n,…, X_kr:n ? X_kr?1:n) is MTP₂ [S-MRR₂] whenever 1 ≤ k₁ < k₂ <···< k_r ≤ n and 1 ≤ r ≤ n. Multivariate likelihood ratio ordering of such general spacing vectors corresponding to two random samples is also considered. These extend some of the results in the literature for usual spacing vectors.     

A One Sided Test Based on Sample Quasi Ranges

Abstract

Let the data from the ith treatment/population follow a distribution with cumulative distribution function (cdf) F _i (x) = F[(x ? μ _i )/θ _i ], i = 1,…, k (k ≥ 2). Here μ _i (?∞ < μ _i  < ∞) is the location parameter, θ _i (θ _i  > 0) is the scale parameter and F(?) is any absolutely continuous cdf, i.e., F _i (?) is a member of location-scale family, i = 1,…, k. In this paper, we propose a class of tests to test the null hypothesis H ₀ ? θ₁ = · = θ _k against the simple ordered alternative H _A  ? θ₁ ≤ · ≤ θ _k with at least one strict inequality. In literature, use of sample quasi range as a measure of dispersion has been advocated for small sample size or sample contaminated by outliers [see David, H. A. (1981). Order Statistics. 2nd ed. New York: John Wiley, Sec. 7.4]. Let X _i1,…, X _in be a random sample of size n from the population π _i and R _ir  = X _i:n?r  ? X _i:r+1, r = 0, 1,…, [n/2] ? 1 be the sample quasi range corresponding to this random sample, where X _i:j represents the jth order statistic in the ith sample, j = 1,…, n; i = 1,…, k and [x] is the greatest integer less than or equal to x. The proposed class of tests, for the general location scale setup, is based on the statistic W _r  = max_1≤i<j≤k (R _jr /R _ir ). The test is reject H ₀ for large values of W _r . The construction of a three-decision procedure and simultaneous one-sided lower confidence bounds for the ratios, θ _j /θ _i , 1 ≤ i < j ≤ k, have also been discussed with the help of the critical constants of the test statistic W _r . Applications of the proposed class of tests to two parameter exponential and uniform probability models have been discussed separately with necessary tables. Comparisons of some members of our class with the tests of Gill and Dhawan [Gill A. N., Dhawan A. K. (1999). A One-sided test for testing homogeneity of scale parameters against ordered alternative. Commun. Stat. – Theory and Methods 28(10):2417–2439] and Kochar and Gupta [Kochar, S. C., Gupta, R. P. (1985). A class of distribution-free tests for testing homogeneity of variances against ordered alternatives. In: Dykstra, R. et al., ed. Proceedings of the Conference on Advances in Order Restricted Statistical Inference at Iowa city. Springer Verlag, pp. 169–183], in terms of simulated power, are also presented.     

The Characterization of the Symmetric Lorenz Curves by Doubly Truncated Mean Function

We characterize symmetric Lorenz curves by the relation m(x, μ²/x) = μ (where μ =E(X) and m(x, y) = E(X | x ≤ X ≤ y) is the doubly truncated mean function). We establish that the points of the r.v. which generate the symmetric points on the Lorenz curve are x and μ²/x, and that all the distribution functions defined on the same support which are generators of the symmetric Lorenz curves have the same mean. We obtain the conditions under which doubly truncated distributions generate symmetrical Lorenz curves.     

Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X₀, X^T, Y^T, Z^T)^T having a multivariate elliptical distribution, we derive the exact joint distribution of (X₀, a^TX_(n), b^TY_[n], c^TZ_[n])^T, where a, b, c∈?ⁿ, X_(n)=(X₍₁₎, …, X_(n))^T, X₍₁₎<···<X_(n), is the vector of order statistics arising from X, and Y_[n]=(Y_[1], …, Y_[n])^T and Z_[n]=(Z_[1], …, Z_[n])^T denote the vectors of concomitants corresponding to X_(n) ((Y_[r], Z_[r])^T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X_(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X₀, X_(r), Y_[r], Z_[r])^T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X_(r), say, based on the concomitants Y_[r] and Z_[r]. Finally, we illustrate the usefulness of our results by a real data.     

Optimal Nonparametric Quantile Estimators. Towards a General Theory. A Survey

“Nonparametric” in the title is used to say that observations X ₁,…,X _n come from an unknown distribution F ∈ ? with ? being the class of all continuous and strictly increasing distribution functions. The problem is to estimate the quantile of a given order q ∈ (0,1) of the distribution F. The class ? of distributions is very large; it is so large that even X _nq:n , where nq is an integer, may be very poor estimator of the qth quantile. To assess the performance of estimators no properties based on moments may be used: expected values of estimators should be replaced by their medians, their variances—by some characteristics of concentration of distributions around the median. If an estimator is median-biased for one of distributions, the bias of the estimator may be infinitely large for other distributions. In the note optimal estimators with respect to various criteria of optimality are presented. The pivotal function F(T) of the estimator T is introduced which enables us to apply the classical statistical approach.     

Information Properties for Concomitants of Order Statistics in Farlie–Gumbel–Morgenstern (FGM) Family

Let (X _i , Y _i ), i = 1, 2,…, n be independent and identically distributed random variables from some continuous bivariate distribution. If X _(r) denotes the rth-order statistic, then the Y's associated with X _(r) denoted by Y _[r] is called the concomitant of the rth-order statistic. In this article, we derive an analytical expression of Shannon entropy for concomitants of order statistics in FGM family. Applying this expression for some well-known distributions of this family, we obtain the exact form of Shannon entropy, some of the information properties, and entropy bounds for concomitants of order statistics. Some comparisons are also made between the entropy of order statistics X _(r) and the entropy of its concomitants Y _[r]. In this family, we show that the mutual information between X _(r) and Y _[r], and Kullback–Leibler distance among the concomitants of order statistics are all distribution-free. Also, we compare the Pearson correlation coefficient between X _(r) and Y _[r] with the mutual information of (X _(r), Y _[r]) for the copula model of FGM family.     

On An Intriguing Distributional Identity

For a continuous random variable X with support equal to (a, b), with c.d.f. F, and g: Ω₁ → Ω₂ a continuous, strictly increasing function, such that Ω₁∩Ω₂?(a, b), but otherwise arbitrary, we establish that the random variables F(X) ? F(g(X)) and F(g^{? 1}(X)) ? F(X) have the same distribution. Further developments, accompanied by illustrations and observations, address as well the equidistribution identity U ? ψ(U) = ^dψ^{? 1}(U) ? U for U ～ U(0, 1), where ψ is a continuous, strictly increasing and onto function, but otherwise arbitrary. Finally, we expand on applications with connections to variance reduction techniques, the discrepancy between distributions, and a risk identity in predictive density estimation.     

Identifiability of distributions under competing risks and complementary risks model

Let X₁,X₂,…,X_p be p random variables with cdf's F₁(x),F₂(x),…,F_p(x)respectively. Let U = min(X₁,X₂,…,X_p) and V = max(X₁,X₂,…,X_p).In this paper we study the problem of uniquely determining and estimating the marginal distributions F₁,F₂,…,F_p given the distribution of U or of V.

First the problem of competing and complementary risks are introduced with examples and the corresponding identification problems are considered when the X₁'s are independently distributed and U(V) is identified, as well as the case when U(V) is not identified. The case when the X₁'s are dependent is considered next. Finally the problem of estimation is considered.     

Kernel Estimators for Distribution Functions on Dependent Random Fields

Consider observations (representing lifelengths) taken on a random field indexed by lattice points. Estimating the distribution function F(x) = P(X _i  ≤ x) is an important problem in survival analysis. We propose to estimate F(x) by kernel estimators, which take into account the smoothness of the distribution function. Under some general mixing conditions, our estimators are shown to be asymptotically unbiased and consistent. In addition, the proposed estimator is shown to be strongly consistent and sharp rates of convergence are obtained.     

Nonparametric Prediction Intervals for Future Order Statistics in a Proportional Hazard Model

Consider k independent random samples with different sample sizes such that the ith sample comes from the cumulative distribution function (cdf) F _i  = 1 ? (1 ? F)^{α _i} , where α _i is a known positive constant and F is an absolutely continuous cdf. Also, suppose that we have observed the maximum and minimum of the first k samples. This article shows how one can construct the nonparametric prediction intervals for the order statistics of the future samples on the basis of these information. Three schemes are studied and in each case exact expressions for the prediction coefficients of prediction intervals are derived. Numerical computations are given for illustrating the results. Also, a comparison study is done while the complete samples are available.     

Attainability of the Upper Bounds for the Mean Past Lifetime of Parallel System Components

Among reliability systems, one of the basic systems is a parallel system. In this article, we consider a parallel system consisting of n identical components with independent lifetimes having a common distribution function F. Under the condition that the system has failed by time t, with t being 100pth percentile of F(t = F ^?1(p), 0 < p < 1), we characterize the probability distributions based on the mean past lifetime of the components of the system. These distributions are described in the form of a specific shape on the left of t and arbitrary continuous function on the right tail.     

Characterizations based on regression assumptions of order statistics

Let X₍₁₎≤X₍₂₎≤···≤X_(n) be the order statistics from independent and identically distributed random variables {X_i, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.     

Bounds on Bivariate Distribution Functions with Given Margins and Known Values at Several Points

Let H(x, y) be a continuous bivariate distribution function with known marginal distribution functions F(x) and G(y). Suppose the values of H are given at several points, H(x _i , y _i ) = θ _i , i = 1, 2,…, n. We first discuss conditions for the existence of a distribution satisfying these conditions, and present a procedure for checking if such a distribution exists. We then consider finding lower and upper bounds for such distributions. These bounds may be used to establish bounds on the values of Spearman's ρ and Kendall's τ. For n = 2, we present necessary and sufficient conditions for existence of such a distribution function and derive best-possible upper and lower bounds for H(x, y). As shown by a counter-example, these bounds need not be proper distribution functions, and we find conditions for these bounds to be (proper) distribution functions. We also present some results for the general case, where the values of H(x, y) are known at more than two points. In view of the simplification in notation, our results are presented in terms of copulas, but they may easily be expressed in terms of distribution functions.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

Locally minimax tests for multiple correlations

Let X be a normally distributed p-dimensional column vector with mean μ and positive definite covariance matrix σ. and let X α, α = 1,…, N, be a random sample of size N from this distribution. Partition X as ( X ₁, X ₍₂₎', X '₍₃₎)', where X₁ is one-dimension, X₍₂₎ is p₂- dimensional, and so 1 + p₁ + p₂ = p. Let ρ₁ and ρ be the multiple correlation coefficients of X₁ with X₍₂₎ and with ( X '₍₂₎, X '₍₃₎)', respectively. Write ρ2/2 = ρ² - ρ2/1. We shall cosider the following two problems     

Some New Results on Likelihood Ratio Orderings for Spacings of Heterogeneous Exponential Random Variables

Let X ₁, X ₂,…, X _n be independent exponential random variables with X _i having failure rate λ _i for i = 1,…, n. Denote by D _i:n  = X _i:n  ? X _i?1:n the ith spacing of the order statistics X _1:n  ≤ X _2:n  ≤ ··· ≤ X _n:n , i = 1,…, n, where X _0:n ≡ 0. It is shown that if λ _n+1 ≤ [≥] λ _k for k = 1,…, n then D _n:n  ≤ _lr D _n+1:n+1 and D _1:n  ≤ _lr D _2:n+1 [D _2:n+1 ≤ _lr D _2:n ], and that if λ _i  + λ _j  ≥ λ _k for all distinct i,j, and k then D _n?1:n  ≤ _lr D _n:n and D _n:n+1 ≤ _lr D _n:n , where ≤ _lr denotes the likelihood ratio order. We also prove that D _1:n  ≤ _lr D _2:n for n ≥ 2 and D _2:3 ≤ _lr D _3:3 for all λ _i 's.     

The Recursive Kernel Distribution Function Estimator Based on Negatively and Positively Associated Sequences

Let {X _j , j ≥ 1} be a strictly stationary negatively or positively associated sequence of real valued random variables with unknown distribution function F(x). On the basis of the random variables {X _j , j ≥ 1}, we propose a smooth recursive kernel-type estimate of F(x), and study asymptotic bias, quadratic-mean consistency and asymptotic normality of the recursive kernel-type estimator under suitable conditions.     

On a Class of Distributions Defined by the Relationship Between Their Density and Distribution Functions

Knowledge concerning the family of univariate continuous distributions with density function f and distribution function F defined through the relation f(x) = F ^α(x)(1 ? F(x))^β, α, β ? , is reviewed and modestly extended. Symmetry, modality, tail behavior, order statistics, shape properties based on the mode, L-moments, and—for the first time—transformations between members of the family are the general properties considered. Fully tractable special cases include all the complementary beta distributions (including uniform, power law and cosine distributions), the logistic, exponential and Pareto distributions, the Student t distribution on 2 degrees of freedom and, newly, the distribution corresponding to α = β = 5/2. The logistic distribution is central to some of the developments of the article.